Atom Interferometer Gyroscope with Spin-Dependent Phase ShiftsInduced by Light near a Tune-Out Wavelength

Raisa Trubko,1 James Greenberg,2 Michael T. St. Germaine,2 Maxwell D. Gregoire,2

William F. Holmgren,2 Ivan Hromada,2 and Alexander D. Cronin1,21College of Optical Sciences, University of Arizona, Tucson, Arizona 85721, USA

2Department of Physics, University of Arizona, Tucson, Arizona 85721, USA(Received 31 October 2014; published 9 April 2015)

Tune-out wavelengths measured with an atom interferometer are sensitive to laboratory rotation ratesbecause of the Sagnac effect, vector polarizability, and dispersion compensation. We observed shifts inmeasured tune-out wavelengths as large as 213 pm with a potassium atom beam interferometer, and weexplore how these shifts can be used for an atom interferometer gyroscope.

DOI: 10.1103/PhysRevLett.114.140404 PACS numbers: 03.75.Dg, 32.10.Dk, 06.30.Gv

Atom interferometers have an impressive variety ofapplications ranging from inertial sensing to measurementsof fundamental constants, measurements of atomic proper-ties, and studies of topological phases [1]. In particular,making a better gyroscope has been a long-standing goal inthe atom optics community because atom interferometershave the potential to outperform optical Sagnac gyroscopes.Advances in the precision and range of applications for atominterferometry have been realized by using interferometerswith multiple atomic species [2–5], multiple atomic veloc-ities [6–10], multiple atomic spin states [11–13], andmultiple atomic path configurations [14–18]. Here, weuse atoms with multiple spin states to demonstrate a newmethod for rotation sensing. Our atom interferometergyroscope shown in Fig. 1 reports the absolute rotation rateΩ in terms of an optical wavelength, using a spin-dependentphase echo induced by light near a tune-out wavelength.A tune-out wavelength, λzero, occurs where the dynamic

polarizability of an atom changes sign between two reso-nances [19–29]. Since atomic vector polarizability dependson spin [30–32], theoretical tune-out wavelengths usuallydescribe atoms with spin mF ¼ 0. The same λzero should befound, on average, for atoms in a uniform distribution of spinstates.However, in this Letter, we show that the Sagnac effectbreaks the symmetry expected from the vector polarizabilityin a way that makes tune-out wavelengths remarkablysensitive to the laboratory rotation rate. We measuredtune-out wavelengths λzero;lab using a potassium atom inter-ferometer and circularly polarized light, and found that ourmeasurements were shifted by 0.213 nm from the theoreticaltune-out wavelength of λzero ¼ 768.971 nm [19]. This shiftismore than 100 times larger than the uncertaintywithwhichλzero can bemeasured [29], and this suggests the possibility ofcreating a sensitive gyroscope using tune-out wavelengths.The purpose of this Letter is therefore to explain how an atominterferometer gyroscope canmeasure the laboratory rotationrate Ω with the aid of atomic spin-dependent phase shiftsinduced by light near a tune-out wavelength. This is a new

application of tune-out wavelengths and a new method foratom interferometry that could improve sensors needed fornavigation, geophysics, and tests of general relativity.Atom interferometer gyroscopes [1,6–9,33–39] can

sense changes in rotation rate (ΔΩ) because of theSagnac effect. Some atom interferometers [7–9,34,35]can also report the absolute rotation rate (Ω) with respectto an inertial frame of reference since the Sagnac phasedepends on atomic velocity. Because the Sagnac phase isdispersive, Ω can affect the interference fringe contrast.References [6–9,34,35] applied auxiliary rotations to anatom interferometer to compensate for the earth’s rotationΩe and thus maximize contrast. References [34,35] evenused contrast as a function of applied rotation rate in order tomeasure Ωe.In comparison, here we demonstrate optical and static

electric field gradients that compensate for dispersion in theSagnac phase. This is a general example of dispersioncompensation [40,41] in which one type of phase compen-sates for dispersion in another. Furthermore, we show thatcircularly polarized light at λzero makes an observableΩ-dependent phase shift for our unpolarized atom beam

atom beam

nanogratings Ωlaser system

x

z

pola

riza

tion

cont

rol

FIG. 1 (color online). Apparatus diagram. The branches of a3-nanograting Mach-Zehnder atom interferometer [1] are illumi-nated asymmetrically by laser light propagating perpendicular tothe page. An optical cavity (not shown) recycles the light toincrease the phase shift. A single mode optical fiber (yellow)guides the laser light into the atom beam vacuum chamber, andthe loops in the fiber are used to control the optical polarization.

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interferometer. This works because spin-dependentdispersion compensation causes higher contrast for one spinstate. Thus, using spin as a degree of freedom and light near atune-out wavelength, we made a gyroscope that reports theabsolute rotation rateΩ in termsof a light-inducedphase shift.Our gyroscope, shown in Fig. 1, uses material nano-

gratings that permit interferometry with distributions ofatomic spin and velocity, both of which are needed in orderto cause the shifts in λzero;lab that are sensitive toΩ. An atominterferometer like ours was previously shown to monitorchanges in rotation rate ΔΩ [33]. We now show that anatom interferometer gyroscope with material nanogratingscan measure absolute rotation rates smaller than Ωe. This issignificant because nanogratings offer some advantagessuch as simplicity, reliability, and spin-independent andnearly velocity-independent diffraction amplitudes thatmay enable more robust and economical Ω sensors.We studied the light-induced phase shift ϕ for an

ensemble of atoms, which we model as

ϕ ¼ ϕon − ϕoff ð1Þ

where ϕon is the measured phase when the light is on andϕoff is the measured phase when the light is off. For an atombeam with a velocity distribution PvðvÞ and a uniformdistribution of spin states PsðF;mFÞ ¼ 1=8, the contrastCon and phase ϕon for the ensemble are described by

Coneiϕon ¼ Co

X

F;mF

PsðF;mFÞZ

∞

0

PvðvÞeiΦtotaldv ð2Þ

where Φtotal¼ΦLþΦSþΦaþΦo. Here, ΦL is the velocity-dependent and spin-dependent phase caused by light, ΦSis the velocity-dependent Sagnac phase, Φa is the velocity-dependent phase induced by an acceleration or gravity,Φo is the initial phase, and Co is the initial contrast ofthe interferometer. A similar equation can be written forCoffeiϕoff with the light off so that ΦL ¼ 0. Our atom beamhas a velocity distribution PvðvÞ adequately described byPvðvÞ ¼ Av3exp½−ðv − v0Þ2=ð2σ2vÞ�, where A is a normali-zation constant [42].The Sagnac phase [33,34]

ΦS ¼4πL2Ωvdg

ð3Þ

is a function of atomic velocity v and the rotation rate Ωalong the normal of the interferometer’s enclosed area. L isthe distance between gratings, and dg is the period of thegratings. In our interferometer, dg ¼ 100 nm andL ¼ 0.94 m, so ΦS ¼ 2.7 rad for a 1600 m=s atom beamin our laboratory at 32° N latitude due to Ωe.The gravity phase Φa [33] is

Φa ¼2πL2g sinðθÞ

v2dgð4Þ

where g sinðθÞ is the gravitational acceleration along thegrating wave vector direction. As we discuss later, θ andΦaare small, but nonzero.The light phase is

ΦL ¼ αðωÞ2ϵocℏv

Zs

�ddx

Iðr;ωÞ�dz ð5Þ

where the dynamic polarizability αðωÞ depends on theatomic state jF;mFi and the laser polarization [30–32].Near the second nanograting, we shine 50 mWof laser lightperpendicular to the plane of the interferometer. The laser’sirradiance gradient in a beam with a 100 μm diameter waistasymmetrically illuminates the atom beam paths assketched in Fig. 1. The irradiance gradient ðd=dxÞI isintegrated along the atom beam paths in the z direction. Thepath separation s is proportional to v−1. Hence, for laserbeams much wider than s, the light phase ΦL approxi-mately depends on v−2. The fact that this does not exactlymatch the v−1 dispersion of the Sagnac phase means thedispersion compensation is imperfect, which is why we seecaustics in Fig. 2. Figure 2 presents modeled phase shiftsfor ground-state potassium atoms with several differentvelocities and five different spin states. Figure 2 illustrateshow spin-dependent dispersion compensation works andhow it can make λzero;lab ≠ λzero.The way ΦS affects the light-induced phase ϕðλÞ leads to

several testable predictions that we experimentally verified.Equation (2) led us to predict a new wavelength λzero;lab forwhich ϕ is zero. A simulation of this prediction is shown in

-6

-5

-4

-3

-2

-1

0

1

phas

e (r

ad)

769.6769.2768.8768.4

wavelength (nm)

mF = 0 mF = 1 mF = -1

mF = 2 mF = -2 ensemble

zero,lab

zero

zero,lab

mF= +2mF= -2

FIG. 2 (color online). Light-induced phase spectra demonstratedispersion compensation. The phase ΦLðλ; vÞ þ ΦSðvÞ − ϕoff isplotted for 95% circularly polarized light interacting with fiveatomic spin states (colors) and a range of atomic velocitiesspanning 80% to 120% of v0 ¼ 2000 m=s. Black curves showspectra for velocity v0 for each spin state. Curves for each spinstate coalesce in caustics at a different λ where spin-dependentΦLðλ; vÞ compensates for dispersion in ΦSðvÞ. The ensemblephase shift (green line) shows the root in ϕ at λzero;lab, which isshifted by −120 pm from λzero. The phasor diagram (inset)illustrates how ΦL compounds with ΦS to increase dispersionfor one spin state and decrease dispersion for another spin state.

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Fig. 2, and data demonstrating þ203 to −213 pm shifts inλzero;lab are shown in Fig. 3.Higher irradiance on the left interferometer path when

looking from the source towards the detector would cause alonger λzero;lab (if the grating tilt were zero so that Φa ¼ 0).This is because attraction towards light on the left compen-sates forΦS in the northern hemisphere, and only spin stateswith roots inαðωÞ at longerwavelengths are attracted to lightat λzero. These states therefore contributewithmoreweight toϕðλzeroÞ because of dispersion compensation. On the otherside, if the irradiance is stronger on the right-hand interfer-ometer path, then repulsion from the light compensates forΦS, and spin states with roots in αðωÞ at shorter wavelengthscontribute more to ϕðλzeroÞ. Grating tilt θ and the gravityphase Φa complicate this picture. In our experiment, thedispersion dΦa=dv is opposite and slightly larger in mag-nitude than the dispersion dΦS=dv, so higher irradiance onthe left path of the atom interferometer causes a shorterλzero;lab. Figure 3 shows data verifying this prediction.We predict that the wavelength difference Δλ ¼

λzero;lab − λzero will not change if the optical k vectorreverses direction, nor if the optical circular polarizationreverses handedness, nor if the magnetic field parallel to theoptical k vector reverses direction. None of these reversalschanges the fact that a potential gradient that is attractivetowards the left side (or repulsive from the right side) isneeded to compensate for the Sagnac phase dispersion inthe northern hemisphere. Therefore, the magnitude jΔλjcan increase if the laser is simply reflected over the atombeam path. We tested this prediction by constructing anoptical cavity with plane mirrors to recycle light so thatthe same interferometer path is exposed to upward and

downward propagating laser beams for several passes. Thisincreased the magnitude of ϕðλzeroÞ as predicted.External magnetic fields also affect λzero;lab. A uniform

magnetic field parallel or antiparallel to the optical kvector maximizes the sensitivity to optical polarization.Alternatively, a magnetic field perpendicular to the opticalk vector reduces Δλ because the atomic spin states precessabout the field so the resulting spin-dependent differencesin light shift time-average to zero. Data in Fig. 4 show thatλzero;lab is closer to λzero when we apply a perpendicularmagnetic field. Residual differences between λzero;lab andλzero are due to imperfect alignment of the magnetic fieldperpendicular to the k vector and the limited (15 G) strengthof the magnetic field.On the basis of the work presented thus far, deducing Ω

from measurements ofΔλ is challenging because it requiresknowing the magnetic field, the laser power, laser polari-zation, laser beamwaist, and the atom beam velocity spread.To solve this problem, we used a static electric field gradientto induce additional phase shifts that mimic the effect ofauxiliary rotation on the atom interferometer (to first orderin v). A measurement of light-induced phase shift as afunction of electric-field-induced phase shift can serve tocalibrate the relationship between Δλ and Ω. Furthermore,we can determine the absolute rotation rate of the laboratoryby measuring the additional phase shift needed to makeλzero;lab ¼ λzero. The phase due to a static electric fieldgradient is

Φ∇E ¼ αð0Þ2ℏv

Zsddx

E2dz ð6Þ

where αð0Þ is the static electric dipole polarizability [5].The observed phase shift for the ensemble of atoms due toan electric field gradient ϕ∇E is calculated using Eq. (2)with Φ∇E added to Φtotal (and ΦL ¼ 0). This phase shift cancompensate for the dispersion in the Sagnac phase uniformlyfor all atomic spin states.In Fig. 5, we show that ϕðλzeroÞ depends continuously on

ϕ∇E, just as Δλ would on Ω. Specifically, ϕðλzeroÞ is thephase shift caused by light at λzero. The data in Fig. 5 are

FIG. 3 (color online). Measured light-induced phase spectraϕðλÞ using elliptically polarized light and a magnetic field parallelto the optical k vector. The open square red data show λzero;lab ¼768.758ð15Þ nm when the laser beam is on the right side of theatom interferometer, and the solid circle blue data show λzero;lab ¼769.174ð7Þ nm when the laser beam is on the left side of the atominterferometer. Each data point is the average of 40 five-s files,and the error bars show the standard error of the mean. Broadband radiation from the tapered amplifier caused a systematicshift of 15(5) mrad that we accounted for in the ϕ data shown.The red and blue curves show the theory using Eqs. (1)–(5) withan additional average over the width of the atom beam. For thesedata, the grating tilt θ was −20ð5Þ mrad.

FIG. 4 (color online). Measured tune-out wavelengths fordifferent orientations of magnetic field and irradiance gradients.Each data point comes from ϕðλÞ spectra such as those shown inFig. 3. For these data, the grating tilt θ was −20ð5Þ mrad.

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obtained by alternately turning∇E on and off, blocking andunblocking the laser, and then repeating the process with anew ∇E strength. Importantly, the root in phase ϕðλzeroÞ atϕroot∇E occurs when the electric field gradient compensatesfor dispersion in ΦS and Φa. We can interpret this conditionmathematically as

ddv

ðΦS þ Φa þ Φ∇EÞ ¼ 0; ð7Þ

and then it becomes unnecessary to know the laser power orto perform the integral over velocity shown in Eq. (2) forreporting Ω. Using the approximation Φ∇E ¼ ϕroot∇E ðv0=vÞ2,we find

Ω ¼ −dgv0ϕroot∇E2πL2

−g sinðθÞ

v0: ð8Þ

Equation (8) does not include ΦL because when Eq. (7) issatisfied there is no net dispersion to break the symmetry;so including ΦLðλzeroÞ in Eq. (2) produces zero ensemblephase shift ϕ. The fact that Eq. (8) does not include ΦL isconvenient because now we can use light at λzero to measureΩ without precise knowledge of the laser spot size,polarization or irradiance, or the resultant slope dϕ=dλ.Those factors affect the precision with which we can findthe root (ϕroot∇E ), but not the value of the root. We alsoemphasize that an electric field gradient can be used toincrease the dynamic range of our gyroscope.To reportΩ, we measured ϕroot∇E ¼ 1.2ð3Þ rad with data in

Fig. 5, we measured v0 ¼ 1585ð10Þ m=s using phasechoppers [43], and we measured θ ¼ −10ð2Þ mrad bycomparing the nanograting bars to a plumb line. We find

Ω ¼ 0.4ð2ÞΩe, which can be compared to the expectedvalue 0.5 Ωe (the vertical projection of Ωe at our latitude of32° N). In Fig. 5, we also show how Coff depends on ϕ∇E.The phase ϕmaxC∇E that maximizes contrast is another way tofind the static electric field gradient that compensatesfor dispersion in the Sagnac phase and acceleration phase.The value of ϕmaxC∇E ¼ 0.6ð2Þ rad leads to Ω ¼ 0.6ð2ÞΩe.The dominant source of error in our experiment was themeasurement of the nanograting tilt. Discrepancy betweenϕmaxC∇E andϕroot∇E indicates a systematic error, possibly causedby de Broglie wave phase front curvature induced by thelaser beam [44], optical pumping, magnetic field gradients,or the broad band component of our laser spectrum.The shot noise limited sensitivity of our atom interfer-

ometer gyroscope can be estimated from the fact thatϕðλzeroÞ changes by 0.22 rad due to 0.53Ωe, and thestatistical phase noise is δϕ ¼ ð2=NÞ1=2½ð1=CoffÞ2þð1=ConÞ2�1=2 which is 0.06 rad=

ffiffiffiffiffiffiHz

pfor Coff ¼ 0.2,

Con ¼ 0.08, and N ¼ ð100 000 counts= secÞ × t. This indi-cates a sensitivity of 0.2Ωe=

ffiffiffiffiffiffiHz

pfor measurements of

rotation with respect to an inertial reference frame, which iscompetitive with methods presented in Refs. [7–9,34,35].To make a more sensitive gyroscope, the scale factor

ϕðλzeroÞ=Ω can be somewhat increased by using more laserpower and a broader velocity distribution. However, a limitto the sensitivity arises from balancing the benefit of anincreased scale factor against the detriment of increasedstatistical phase noise. This compromise occurs becausemaximizing the scale factor ϕðλzeroÞ=Ω requires significantcontrast loss from the two mechanisms described byEq. (2): first, averaging over the spread in ΦS (which isaffected by σv) and second, averaging over the distributionin ΦL (which is affected by the laser power and polariza-tion). Optimizing σv and laser power can increase thesensitivity (for the same flux and contrast) to 0.05Ωe=

ffiffiffiffiffiffiHz

pfor Ω measurements.This work also indicates how to make measurements of

λzero more independent of Ω. Experiments are less sensitiveto Ω if they use linearly polarized light, a narrow velocitydistribution, a perpendicular magnetic field, and an addi-tional dispersive phase such as Φ∇E to compensate for ΦS.For example, the λzero measurements in Ref. [29] were notsignificantly affected by Ω because there was minimalcontrast loss at λzero. Specifically, the sharp velocitydistribution ðv0=σv ¼ 18Þ caused dispersion in ΦS þ Φathat reduced Coff by less than 1% of C0, and ΦL reducedCon by 4% of C0; so shifts in λzero;lab were less than 1pm inRef. [29]. To increase sensitivity to Ω for measurementsreported here, in Figs. 3–5 we used a broad velocitydistribution (v0=σv ¼ 7) so ΦS þ Φa reduced Coff by 8%of C0, and we also used a large irradiance gradient withcircular polarization that reduced Con by 40% of C0.In summary, an atom beam interferometer with multiple

atomic spin states enabled us to demonstrate systematicshifts in tune-out wavelength measurements (λzero;lab) that

FIG. 5 (color online). (top) Contrast data as a function of phaseshift induced by an electric field gradient ϕ∇E. A Gaussian fit(dashed black line) to the red data points shows that a maximumin contrast occurs at ϕ∇E ¼ 0.6ð2Þ rad due to dispersioncompensation. The solid red curve shows the theory usingEqs. (1)–(6) with Ω ¼ 0.6Ωe. (bottom) Light-induced phase shiftϕ as a function of ϕ∇E, using light at λzero ¼ 768.971 nm. Anerror function fit (dashed black line) to the blue data points showsthe root ϕroot∇E ¼ 1.2ð3Þ rad. The solid blue curve shows the theoryusing Eqs. (1)–(6) with Ω ¼ 0.4Ωe. For these data, the grating tiltθ was −10ð2Þ mrad. The solid green curves show contrast andphase theory for Ω ¼ 0, but the same θ ¼ −10 mrad.

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are larger than 200 pm due to rotation and acceleration.Then, we used the phase induced by light at a theoreticaltune-out wavelength ϕðλzeroÞ as a function of an additionaldispersive phase ϕ∇E applied to report the rotation rate ofthe laboratory with an uncertainty of 0.2Ωe. This work is anew application for tune-out wavelengths, paves the wayfor improving precision measurements of tune-out wave-lengths, and demonstrates a new technique for atominterferometer gyroscopes. The spin-multiplexing tech-niques demonstrated here may find uses in other atom[12,13] and neutron [45,46] interferometry experiments,NMR gyroscopes, and NMR spectroscopy.

This work is supported by NSF Grant No. 1306308 and aNIST PMG. Authors R. T. and M. D. G. also thank NSFGRFP Grant No. DGE-1143953 for support. We thankProfessor Brian P. Anderson for helpful discussions.

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